Questions on the use of algebraic techniques to prove geometric theorems.

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Changing the side of a triangle without changing area?

$\triangle ABC$ has vertices $A=(8,2)$, $B=(0,6)$ and $C=(-3,2)$. Point $C$ can be moved along a certain line with points $A$ and $B$ remaining stationary so that the area of $ABC$ will not change? ...
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1answer
31 views

How to calculate a point between two angled lines based on distance from the lines?

Please take a look at the picture below for the diagram reference: I am trying to calculate the point where it is perfectly 3.3 cm vertically from the 44.52 cm line AND 5.5 cm horizontally from the ...
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67 views

Show that the co-ordinates of the point on the join of $(-3, 7, -13)$ and $(-6, 1, -10)$ which is nearest to the intersection of the planes

Show that the co-ordinates of the point on the join of $(-3, 7, -13)$ and $(-6, 1, -10)$ which is nearest to the intersection of the planes $3x-y- 3z + 32 =0$ and $3x+2y-15z= 8$ is $(-7,-1,-9)$. ...
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19 views

Verify that $R_{(a,b)}\subset D$.

Let $(a,b)$ be any point in the disk $D = \{(x,y): x^2 + y^2 < 1\}$. Put $r=\sqrt{a^2 + b^2}$. Let $R_{(a,b)}$ be the open rectangle with vertices at the points $\left(a\pm\frac{1-r}{8}, b \pm\frac{...
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1answer
33 views

Find the co-ordinates of the point on the join of two points which is nearest to the intersection of two planes

Find the co-ordinates of the point on the join of $(-3, 7, -13)$ and $(-6, 1, -10)$ which is nearest to the intersection of the planes $3x-y- 3z + 32 =0$ and $3x+2y-15z= 8$. Please give me an ...
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1answer
27 views

What are the coordinates for the center of the second circle? (Full question in body)

Full Question:A circle has its center at (6,7) and goes through the point (1,4). A second circle is tangent to the first circle at the point (1,4) and has one-fourth the area. What are the coordinates ...
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2answers
68 views

What is condition for second degree equation to represent a pair of straight lines?

According to my text the necessary and sufficient condition for a general equation of second degree i.e. $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ to represent a pair of straight lines is that 1) the ...
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1answer
18 views

Find the coordinates of E, G and H, and calculate the area of shape OFEH

Currently I am looking at a graph of a circle. The diameter is y=2x+3 Tangent at point E cuts the x-axis at F (12;0) 1. find the coordinates of E 2. find the coordinates of G and H (H being the centre)...
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calculate the value of P is the points A(6;5), B(3;2) and C (2p;p+4) are co-linnear

Also honestly have no clue whatsoever. I have tried jotting down a graph and just finding the differences between A and B and minusing them from B to create C. I know this is completely wrong! Please ...
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In a triangle $ABC$ with $A=(1,3) ,B =(q,0), C =(p,-4)$ [closed]

Let $A=(1,3),B =(q,0), C =(p,-4)$, with $p>0$, the slope of $AB$ is $+45^\circ$ and $AC= \sqrt{50}$. Determine the gradient of $AB$ Calculate the equation of the line $AB$ Calculate the value of ...
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53 views

Find the equations of the lines of greatest slope and least slope

Find the equations of the lines of greatest slope and least slope on the plane $3x-4y+5z-5=0$ drawn through the point $(1,2,2)$ given that the plane $4x-5y+6z-6=0$ is horizontal. I do not need the ...
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3answers
83 views

Find minimum of $a+b$ under the condition $\frac{m^2}{a^2}+\frac{n^2}{b^2}=1$ where $m,n$ are fixed arguments

Assume $m,n \in \mathbb{R}$ is fixed. And $a,b(a>b>0)$ satisfied the equation $$\frac{m^2}{a^2}+\frac{n^2}{b^2}=1$$ Find $\min\{a+b\}$
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Given three coordinates (a,b,c), (d,e,f), and (l,m,n), what is the center of the circle in the 3D plane (h,k,i) that contains these three points.

I have tried the following: $$(a-h)^2+(b-k)^2+(c-i)^2=r^2$$ $$(d-h)^2+(e-k)^2+(f-i)^2=r^2$$ $$(l-h)^2+(m-k)^2+(n-i)^2=r^2$$ Subtracted equation 2 from 1, equation 3 from equation 2, and equation 3 ...
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31 views

Proving a vector bisect two other vectors

How can I prove the vector: $$ \vec{w}=|\vec{u}|\vec{v} + |\vec{v}| \vec{u} $$ bisects the angle between the vectors $\vec{u}$ and $\vec{v}$ ? I have trying using the scalar product, but it does not ...
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17 views

Understanding distance between point and line via infimum

The distance between point and line independently on metric is defined by $$d(X, l) = \inf\{d(X, Y)|Y\in l\}.$$ I have troubles understandning how this infimum works. Can someone please give me an ...
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2answers
49 views

How can I solve this line & plane intersect question and verify the given answer? [closed]

Find an equation for the plane that passes through the point $(3,2,1)$ and contains the line of intersection of the planes with equations $x+y+z=3$ and $x+2y+3z=6$. The given answer from the key is: $...
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1answer
56 views

How to find whether equation of angle bisector represents the obtuse or acute angle bisector of two given straight lines?

Two lines: $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ are given. I know that the equation of its bisectors is ${a_1x + b_1y + c_1 \over \sqrt{(a_1^2 + b_1^2)}} = \pm {a_2x + b_2y + c_2 \over\...
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1answer
48 views

When does $ax+by+c=0$ represents a family of straight lines passing through a fixed point?

a first degree linear equation $ax+by+c=0$ represents a family of straight lines passing through a fixed point if and only if there is linear relationship between a,b and c? How can we prove this? ...
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Equation of a Plane

I realize this may be VERY low level for this forum. I'm practicing for an exam and I just want to verify an answer because I do not have the solutions for this practice test. The question is: Find ...
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3answers
32 views

PARABOLA : Problem

Find the equation of line touching both the parabolas $$ x^2=-32y.......(1)$$ $$ y^2=4x.........(2) $$ i have equated slopes of both the parabolas and applied the condition that all the points on ...
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1answer
18 views

How are trigonometric ratios function of interior angles in a right angled triangle?

How can one assume that the ratio altitude/hypotenuse is a function of angle. For a general right-angled triangle--->Let: Hypotenuse$=c$ Altitude$=a$ Base$=b$ and angle opposite ...
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31 views

Find the radius of the circle for given conditions

A circle with center at origin passes through three points $P$, $Q$ and $R$ with the line segment $PQ$ as its diameter along $x$-axis. A line passes through $P$ intersects the chord $QR$ at point $D$. ...
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1answer
19 views

Find the gradient of lines joining the following pair of points.

If, $Gradient = \frac{(y_2-y_1)}{(x_2-x_1)}$ And, $(x_1,y_1),(x_2,y_2) = (p+3, p-3), (3p+4, p-5)$ Then, $(y_2,y_1) = ((p-5)-(p-3))$ $=((p-5)-p+3)$ $=(p-5-p+3)$ $=(-2)$ And, $(x_2,x_1) = ((...
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48 views

Ring of germs of holomorphic functions at $0\in \mathbb{C}$

So I've been reading the book and they used a induction proof where they just state that for the base case the ring of germs of holomorphic functions on $\mathbb{C}$ is Noetherian. I looked at other ...
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1answer
18 views

Perturbation of tangent ball

As picture below, $A$ and $B$ are two balls, $\partial A\bigcap \partial B=\{k\}$, and $B$ contains $A$. How to show that $$ \forall h\in \partial B,\exists ~\varepsilon > 0 ~st~ A\subset B+\...
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1answer
27 views

if I know a point nearest the zero of a polynomial can I tell which zero it is? (finding intersect of $f(x)$ with a line)

I have a function $f(x)$ and two points $p_1$ and $p_2$. What I need to find is the point where $f(x)$ and the line defined by the two point intersect. I know what $f(x)$ is, $f(x) =\dfrac{c_0+c_2x+...
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1answer
33 views

Question about the Jacobian of a function

Let $f:U\rightarrow V$ , $U$ and $V$ open subsets of $\mathbb{R}^2$, be a smooth function. Let $Jf_p$ be the jacobian of $f$ in the point $p\in U$ and set $M_p:=\sup\{|df_pv|:\|v\|=1\}$ and $m_p:=\inf\...
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Requesting formulas about circum-circles

With the co-ordinates of the three vertices of a triangle given, we have nice looking formulas for the centroid and in-center. Do we have the same kind of formula(s) for the circum-circle or circum-...
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3answers
48 views

Why the Cartesian equations, are called “Cartesian”?

I've been studying analytic geometry and I'm wondering "Why the Cartesian equations, are called 'Cartesian,'" I know that the name is from the René Descartes philosopher. But in that one case why is ...
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1answer
18 views

Equation of a subspace given basis

Suppose we have a subspace expressed as the linear combination of two vectors (basis): $S = x * (3, -3, 1) + y * (5, 1, 3)$ How can I find the equation for the subspace (in this case, a plane ...
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1answer
77 views

A parallelogram between two points on a hexagonal lattice containing all the shortest paths

For any two points on a hexagonal grid with integer coordinates there is a unique parallelogram which contains all of the shortest paths (in terms of taxicab norm) between these points. See the ...
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1answer
22 views

Number of lattice points in triangle formed by x-axis, y-axis and given line

Given a line $ax+by=c$ where $a,b,c$ are positive integers. Is there any formula to find the number of points inside the triangle formed by this line, $x$-axis and $y$-axis? Points on the boundary ...
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2answers
35 views

Find closest Point to Another Point

How do I find the closest point to $(2,2)$ on the line CD, if C is $(3,2)$ and D is $(5, 3)$? How would I solve using linear algebra? Does it involve cross product and distance? Not sure how to solve
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Cohomology of $\mathcal{O}^*$ and projection map

Suppose $X$ is a complex manifold and $T$ a complex space (or complex manifold maybe) and let $\pi:T\times X \rightarrow T$ denote the projection. What are sufficient conditions on $X$ that make $$H^2(...
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1answer
28 views

2D coordinates of rotating a “bent line”?

I have this problem, when I am given a point A an an XY plane, and I need to find the coordinates of a point B that is of a constant distance of my point A, and my OAB angle is fixed (O being the ...
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1answer
26 views

A question about affine spaces

Are there affine spaces that contain subsets that aren't closed to affine combinations of three points? This is a surprising question. I think that exists that kind of affine spaces,but I don't know ...
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33 views

$P$ is a point on ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ $(a>b)$ and $S$ and $S'$ are its focii

If $P$ is a point on ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ $(a>b)$ and $S$ and $S'$ are its focii. $\angle PSS'=\alpha$ and $\angle PS'S=\beta$, then prove that: $$ \tan\left(\frac{\alpha}{...
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2answers
54 views

Best Fitting Pipe in parabolic trench

A work crew is digging a pipeline. The cross section of the trench is in the shape of the parabola $y = x^2$. The pipe has a circular cross section. If the pipe is too large, then the pipe will not ...
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1answer
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find the coordinates of the point that divides the join of A(-1,-7) & B(1,2) internally, in 2:1.

What I wanted to ask was that after finding the coordinates of the point my answer was (1/3, -1) now since the ordinate is -ve doesn't that make this an external division? How can it divide the line ...
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4answers
56 views

Coordinate Geometry: Are there enough information to find out the coordinates?

Question: Given the circle $x^2+y^2=25$ is inscribed in triangle $\triangle ABC$, where vertex $B$ lies on the first quadrant. Slope of $AB$ is $\sqrt 3$ and has a positive y-coordinate, and $|AB|=|AC|...
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2answers
29 views

Rotation of conics [duplicate]

How to rotate a conic by an determined angle? Could someone give me the step by step? (I know how to rotate the coordinate system by that formula \begin{align} x &= x'\cos(a) - y'\sin(a) \\ y &...
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1answer
59 views

A mirror focusing beams at one point

How can I find a shape of a mirror which focuses all parallel beams in one point? I tried to do it in this way: The mirror must be symmetric hence I assumed it has a center in the point $(0,0)$. The ...
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2answers
74 views

Get the four corners of a rectangle

I have a boundary given ($xMin$, $yMin$, $xMax$, $yMax$) and the two points of a reference line of a rectangle. The begin point is at $(x_b, y_b)$ and the end point is at $(x_e, y_e)$. This reference ...
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2answers
25 views

Polar Equation to Rectangular

$$r=\frac{9}{4 \cos θ − 3 \sin θ}$$ How do I do this? (Equation is in polar form.) I have already tried to do this, but I don't know how to finish it.
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Give a geometrical interpretation of the intersection of the planes with equations [closed]

Give a geometrical interpretation of the intersection of the planes with equations \begin{align} &x + y − 3 = 0\\ &y + z + 5 = 0\\ &x + z + 2 = 0 \end{align} what is a geometrical ...
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3answers
28 views

Through the point $A(4,5)$ a line is drawn.

Through the point $A(4,5)$ a line is drawn inclined at $45°$ with the $+ve$ X - axis. It meets $x+y=6$ at the point $B$. Find the equation of $AB$. My solution.. Equation of $AB$ $$(y-y_1)=m(x-x_1)...
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2answers
65 views

Curve equidistant to sine and cosine.

If I have the sine and cosine curves plotted, what would be the formula of the curve that is equidistant to both curves? Here's a picture of how it looks like. The original question comes from a ...
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2answers
76 views

Geometrical interpretation of solving a $3 \times 3$ system of equations

Solve the following system of equations and give a geometrical interpretation of the result. \begin{align*} x + y + z &= 6\\ 2x + y − 3z &= -5\\ 4x − 5y + z &= −3 \end{align*} I know that ...
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1answer
38 views

Plane $3x + y - z= 4$ touches the ellipsoid $2z^2 = \sqrt7(1 - 2x^2 -y^2)$

Show that the Plane $3x + y - z= 4$ touches the ellipsoid $2z^2 = \sqrt7(1 - 2x^2 -y^2)$ My attempt: First I tried to convert the equation of ellipsoid in general form and then further applying the ...
2
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1answer
119 views

condition for cones to be reciprocal

Question : Show that the cone $$ax^2 + by^2 + cz^2 - cxy - ayz - bzx = 0$$ is the reciprocal of the cone $$(a^2 - bc)x^2 + (b^2 - ac)y^2 + (c^2 - ab)z^2 - 2(a^2 + bc)yz - 2(b^2 + ac)zx - 2(c^2 + ab)xy ...